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Submitted on 13 Nov 2018
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A small solid body with large density in a planar fluid is
negligible
Jiao He, Dragos Iftimie
To cite this version:
A SMALL SOLID BODY WITH LARGE DENSITY IN A PLANAR FLUID IS NEGLIGIBLE
JIAO HE AND DRAGOS, IFTIMIE
Abstract. In this article, we consider a small rigid body moving in a viscous fluid filling the whole R2. We assume that the diameter of the rigid body goes to 0, that the initial
velocity has bounded energy and that the density of the rigid body goes to infinity. We prove that the rigid body has no influence on the limit equation by showing convergence of the solutions towards a solution of the Navier-Stokes equations in the full plane R2.
1. Introduction
In this paper, we consider a fluid-solid system consisting in a small smooth rigid body Ωε of size ε evolving in a viscous fluid filling the whole of R2. Our aim is to determine the
limit of this coupled system when the size of the rigid body ε goes to 0.
Let us describe now the fluid solid system of equations. To do that, we need to introduce some notation. We denote by uε, respectively pε, the velocity, respectively the pressure,
of the fluid; they are defined on R2\ Ω
ε, the exterior of the smooth rigid body Ωε. The
evolution of the rigid body Ωε(t) is described by hε, the position of its center of mass, and
by θε, the angle of rotation of the rigid body compared with the initial position. We have
that Ωε(t) = hε(t) + cos θε(t) − sin θε(t) sin θε(t) cos θε(t) Ωε(0) − hε(0).
The velocity of the fluid verifies the incompressible Navier-Stokes equations in the ex-terior of the rigid body:
(1) ∂uε
∂t + uε· ∇uε− ν∆uε+ ∇pε= 0, div uε= 0 for t > 0 and x ∈ R
2\ Ω
ε(t).
On the boundary of the rigid body we assume no-slip boundary conditions: (2) uε(t, x) = h0ε(t) + θ
0
ε(t)(x − hε(t))⊥for t > 0 and x ∈ ∂Ωε(t).
Moreover, the velocity is assumed to vanish at infinity:
(3) lim
|x|→∞uε(t, x) = 0 for t ≥ 0.
Now we write down the equations of motion of the solid body. Let us denote by mε
the mass of the solid and by Jε the momentum of inertia of the solid. We also denote by
σ(uε, pε) the stress tensor of the fluid:
σ(uε, pε) = 2νD(uε) − pεI2
where I2 is the identity matrix and D(uε) is the deformation tensor
Then the solid body Ωε(t) evolves according to Newton’s balance law for linear and angular momenta: (4) mεh00ε(t) = − Z ∂Ωε(t) σ(uε, pε)nε for t > 0, and (5) Jεθ00ε(t) = − Z ∂Ωε(t) (σ(uε, pε)nε) · (x − hε)⊥for t > 0.
Above nε denotes the unit normal to ∂Ωεwhich points to the interior of the rigid body
Ωε, the orthogonal x⊥ is defined by x⊥ = (−x2, x1) and σ(uε, pε)nε denotes the matrix
σ(uε, pε) applied to the vector nε.
One can obtain energy estimates for this system of equations. If we formally multiply the equation of uεby uε, do some integrations by parts using also the equations of motion
of the rigid body, we get the following energy estimate: (6) kuε(t)k2L2(R2\Ω ε)+ mε|h 0 ε(t)|2+ Jε|θ0ε(t)|2+ 4ν Z t 0 kD(uε)k2L2(R2\Ω ε) ≤ kuε(0)k2L2(R2\Ω ε)+ mε|h 0 ε(0)|2+ Jε|θ0ε(0)|2.
To solve the system of equations (1)–(5), we need to impose the initial data. For the fluid part of the system we need to impose the initial velocity uε(0, x). The two equations
describing the evolution of the rigid body are second-order in time, so we need to know hε(0), h0ε(0), θε(0) and θε0(0). The system of equations being translation invariant, we can
assume without loss of generality that the initial position of the center of mass of the rigid body is in the origin: hε(0) = 0. Moreover, from the definition of the angle of rotation θε
we obviously have that θε(0) = 0. So we only need to impose uε(0, x), h0ε(0) and θε0(0).
The initial velocity will be assumed to be square integrable only. As such, its trace on the boundary is not well-defined. Only its normal trace is defined thanks to the divergence free condition. Therefore, we need to impose the following compatibility condition on the initial velocity:
(7) uε(0, x) · nε =h0ε(0) + θε0(0) x − hε(0)
⊥
· nε on ∂Ωε(0).
In conclusion, to solve the system of equations (1)–(5), we need to impose that uε(0, x) ∈
L2(R2\ Ωε(0)), that div uε(0, x) = 0 in R2 \ Ωε(0) and the compatibility condition (7).
There is no condition required on h0ε(0) and θ0ε(0) while hε(0) = 0 and θε(0) = 0.
To state the classical result of existence and uniqueness of solutions of (1)–(5), it is practical to extend the velocity field uε inside the rigid body as follows:
(8) euε(t, x) =
(
uε(t, x) if x ∈ R2\ Ωε(t)
h0ε(t) + θε0(t)(x − hε(t))⊥ if x ∈ Ωε(t).
The conditions imposed on the initial data ensure that ueε(0, x) belongs to L2(R2) and
is divergence free in R2.
Let us denote by ρε the density of the rigid body Ωε. We extend ρε in the fluid region
Due to the energy estimates (6), global existence of finite energy solutions of (1)–(5) have been proved in a variety of settings. The literature is vast, we give here just a few references dealing with the dimension two: in [2], [7] and [13] the authors consider the case of one or several rigid bodies moving in a bounded domain filled with a viscous fluid while in [15] the authors consider a single disk moving in a fluid filling the whole plane. The existence for the problem we are considering here was not explicitly studied in these works (because we do not assume the rigid body to be a disk), but more complicated cases have been considered in the literature: the case of a 2D bounded domain where collisions with the boundary must be taken into account (see [2], [7] and [13]) and the case of R3 with a rigid body of arbitrary shape (see for example [16] and [14]). From these results we can extract the following statement about the existence and uniqueness of solutions of (1)–(5). We use the notation R+= [0, ∞) and emphasize that the endpoint
0 belongs to R+. This is important when we write local spaces in R+ like for instance
L2loc(R+) = {f ; f square integrable on any interval [0, t]}. We will give a formulation of
the PDE in terms of the extended velocity ueε.
Theorem 1. Let uε(0, x) ∈ L2(R2\Ωε(0)) be divergence free and verifying the compatibility
condition (7). We assume that hε(0) = 0 and θε(0) = 0 and we extend uε(0, x) to euε(0, x) as in (8). Then euε(0, x) is divergence free and square integrable on R2 and there exists a
unique global weak solution (uε, hε, θε) of (1)–(5) in the following sense:
• uε, hε, θε satisfy
uε ∈ L∞(R+; L2(R2\ Ωε)) ∩ L2loc(R+; H1(R2\ Ωε)),
hε ∈ W1,∞(R+; R2), θε∈ W1,∞(R+; R);
• if we define euε as in (8) then ueε is divergence free with Dueε(t, x) = 0 in Ωε(t) and the equations of motion are verified in the sense of distributions under the following form − Z ∞ 0 Z R2 e ρεueε· ∂tϕε+ (ueε· ∇)ϕε + 2ν Z ∞ 0 Z R2 D(ueε) : D(ϕε) = Z R2 e ρε(0)ueε(0) · ϕε(0). for any divergence free test function ϕε ∈ H1(R+× R2) compactly supported in
time and such that Dϕε(t, x) = 0 in Ωε(t);
Moreover, ueε satisfies the following energy inequality: (9) Z R2 e ρε|ueε| 2+ 4ν Z t 0 Z R2 |D(ueε)|2 ≤ Z R2 e ρε(0)|ueε(0)| 2 ∀t > 0.
As mentioned before, we are interested in describing the asymptotic behavior of this fluid-solid system when the diameter of the rigid body Ωε goes to 0. There are several
papers dealing with this issue when the rigid body does not move with the fluid. Iftimie, Lopes Filho and Nussenzveig Lopes [9] have treated the asymptotic behavior of viscous incompressible 2D flow in the exterior of a small fixed rigid body as the size of the rigid body becomes very small, see also [1] for the case of the periodic boundary conditions. Moreover, Lacave [10] considered a two-dimensional viscous fluid in the exterior of a thin fixed rigid body shrinking to a curve and proved convergence to a solution of the Navier-Stokes equations in the exterior of a curve.
Although we are dealing here only with viscous fluids, let us mention that the case of a perfect incompressible fluid governed by the Euler equations also makes sense and the literature is richer. Let us mention a few results. Iftimie, Lopes Filho and Nussenzveig Lopes [8] have studied the asymptotic behavior of incompressible, ideal two-dimensional
flow in the exterior of a small fixed rigid body when the size of the rigid body becomes very small. Recently, Glass, Lacave and Sueur [4] have studied the case when the solid body shrinks to a point with fixed mass and circulation and is moving with the fluid. The same three authors also consider in [5] the case when the body shrinks to a massless pointwise particle with fixed circulation. In that case, the fluid-solid system converges to the vortex-wave system. In addition, Glass, Munnier and Sueur [6] considered the case of a bounded domain.
As far as we know, there is only one result dealing with the case of a small rigid body moving in a viscous fluid in dimension two. More precisely, Lacave and Takahashi [11] considered a small moving disk in a two-dimensional viscous incompressible fluid. They used a fixed-point type argument based on previously known Lp−Lqdecay estimates of the
linear semigroup associated to the fluid-solid system (see [3]). They proved convergence towards the solution of the Navier-Stokes equations in R2 under the assumption that the rigid body is a disk of radius ε, that the density ρε is constant plus some smallness
assumptions on the initial data (including the smallness of the L2 norm of the initial fluid velocity). More precisely, their result is the following.
Theorem 2 ([11]). There exists a constant λ0> 0 such that if
• uε(0, x) ∈ L2(R2\ Ω
ε(0)) is divergence free and verifies the compatibility condition
(7);
• the rigid body is the disk Ωε= D(hε, ε);
• the density ρε is assumed to be independent of ε;
• euε(0, x) converges weakly in L2(R2) to some u0(x);
• we have the following smallness of the initial data (10) kuε(0, x)kL2(R2\Ω ε(0))+ ε|h 0 ε(0)| + ε2|θ 0 ε(0)| ≤ λ0
then the global solution euε given by Theorem 1 converges weak∗ in L
∞
(R+; L2(R2)) ∩
L2loc(R+; H1(R2)) towards the weak solution of the Navier-Stokes equations in R2 with
initial data u0.
Although they state their result for constant density, presumably the proof can be adapted to the case where ρε≥ ρ0 for some ρ0 > 0 independent of ε. On the other hand,
the hypothesis that Ωε is a disk seems to be essential in the result of [11]. Indeed, a key
ingredient are the estimates of [3] and the proof of that result relies heavily on the fact that Ωε is a disk because it uses explicit formulae valid only for the case of a disk. Moreover, it
is also hard to see how the smallness condition (10) could be removed in their argument. Indeed, they use a fixed point argument and that requires smallness at some point. Let us observe that in [11] the authors also obtain uniform bounds in ε for the velocity of the disk. Therefore, they can prove that the center of mass of the disk converges to some trajectory. However, nothing can be said about this limit trajectory.
Here, we improve the result of [11] in two respects. First, the rigid body does not need to be a disk. It does not even need to be shrinking homothetically to a point like in [11]. We only assume that the diameter of the rigid body goes to 0. Second, we require no smallness assumption on the initial fluid velocity uε(0, x). On the other hand, we need to
assume that the density of the rigid body goes to infinity and we are not able to prove uniform bounds on the motion of the rigid body as in [11]. More precisely, we will prove the following result.
Theorem 3. We assume the hypothesis of Theorem 1 and moreover • Ωε(0) ⊂ D(0, ε);
• the mass mε of the rigid body verifies that
(11) mε
ε2 → ∞ as ε → 0;
• uε(0, x) is bounded independently of ε in L2(R2\Ωε(0)) and√mεh0ε(0) and
√ Jεθ0ε(0)
are bounded independently of ε;
• euε(0, x) converges weakly in L2(R2) to some u0(x) where euε(0, x) is constructed as in (8).
Let (uε, hε, θε) be the global solution of the system (1)–(5) given by Theorem 1. Then ueε converges weak∗ in L∞(R+; L2(R2)) ∩ L2loc(R+; H1(R2)) as ε → 0 towards the solution of
the Navier-Stokes equations in R2 with initial data u0.
It will be clear from the proof that the convergence of euε is stronger than stated. For
instance, we shall prove that euε converges strongly in L
2
loc (see Section 4).
Let us remark that if the measure of Ωε is of order ε2 (something which is true if the
rigid body shrinks homothetically to a point, i.e. if Ωε(0) is ε times a fixed rigid body)
then the hypothesis (11) means that the density ρε of the rigid body goes to ∞ as ε → 0.
Observe next that the boundedness of uε(0, x) in L2(R2\ Ωε(0)) and the boundedness
of √mεh0ε(0) and
√
Jεθε0(0) imply the boundedness of
p e
ρε(0, x)ueε(0, x) in L
2(R2). Since
ρε→ ∞ this implies thatueε(0, x) is bounded in L
2(R2). Therefore, the weak convergence
of ueε(0, x) to u0(x) is not really a new hypothesis. Moreover, the boundedness of pρeε(0, x)ueε(0, x) in L
2(R2) and the energy inequality
(9) imply that pρeεeuε is bounded independently of ε in the space L
∞
(R+; L2(R2)) ∩
L2
loc(R+; H1(R2)). Using again that ρε → ∞ we deduce thatueε is also bounded indepen-dently of ε in L∞(R+; L2(R2)) ∩ L2loc(R+; H1(R2)). And this is all we need to prove the
convergence ofueε towards a solution of the Navier-Stokes equations in R
2. Our proof does
not require that ueε verifies the boundary conditions on Ωε, nor do we need that Dueε = 0 in Ωε. We only need the above mentioned boundedness of euε and the fact that it verifies the Navier-Stokes equations (without any boundary condition) in the exterior of the disk D(hε(t), ε). We state next a more general result.
Theorem 4. Let vε be a time-dependent divergence free vector field defined on R+× R2
belonging to the space
(12) L∞(R+; L2(R2)) ∩ L2loc(R+; H1(R2)) ∩ Cw0(R+; L2loc(R2\ D(hε(t), ε)))
and let hε∈ W1,∞(R+; R2). Assume moreover that
• vε is bounded independently of ε in the above space;
• vε(0, x) converges weakly in L2 as ε → 0 to some v0(x);
• vε verifies the Navier-Stokes equations in the exterior of the disk D(hε(t), ε):
(13) ∂tvε− ν∆vε+ vε· ∇vε= −∇πε in the set {(t, x) ; t > 0 and |x − hε(t)| > ε}
for some πε;
• the velocity of the center of the disk verifies that ε|h0ε(t)| → 0 in L∞loc(R+) when
ε → 0.
Let v be the unique solution of the Navier-Stokes equations in R2 with initial data v0.
Then vε converges to v as ε → 0 weak∗ in the space L∞(R+; L2(R2)) ∩ L2loc(R+; H1(R2)).
Theorem 4 with vε = euε implies Theorem 3. Indeed, we already observed above that e
uε has all the properties required from vε in Theorem 4. And the hypothesis made on
the mass of the rigid body, see relation (11), in Theorem 3 implies that ε|h0ε(t)| → 0 in
L∞loc(R+) when ε → 0. This can be easily seen from the energy estimate (6). Indeed, the
hypothesis of Theorem 3 implies that the right-hand side of (6) is bounded uniformly in ε so √mεh0ε is uniformly bounded in t and ε. The fact that mε2ε → ∞ and the boundedness
of √mεh0ε implies that εh0ε→ 0 as ε → 0 uniformly in time.
The idea of the proof of Theorem 4 is completely different from the proof given in [11]. We multiply (13) with a cut-off vanishing on the disk D(hε(t), ε) constructed in a very
particular manner. We then pass to the limit with classical compactness methods. The difficulty here is that the cut-off function itself depends on the time, so time-derivative estimates of vε are not so easy to obtain. Also, passing to the limit in the terms ∂tv and
∆v is not obvious: the first is difficult because the time derivative is hard to control and the second one is difficult because the cut-off introduces negative powers of ε in this term. The plan of the paper is the following. In the following section we introduce some notation and prove some preliminary results. In Section 3 we construct the special cut-off near the rigid body. The required temporal estimates are proved in Section 4. Finally, we pass to the limit in Section 5.
2. Notation and preliminary results
We use the classical notation Cm for functions with m continuous derivatives and Hm the Sobolev space of functions with m square-integrable weak derivatives. The notation Cbm stands for functions in Cm with bounded derivatives up to order m. All function spaces and norms are considered to be taken on R2 in the x variable unless otherwise specified. We define C0,σ∞ to be the space of smooth, compactly supported and divergence free vector fields on R2. The derivatives are always taken with respect to the variable
x unless otherwise specified. The double dot product of two matrices M = (mij) and
N = (nij) denotes the quantity M : N =Pi,jmijnij. We denote by C a generic universal
constant whose value can change from one line to another.
Let ϕ ∈ Cb1(R+; C0,σ∞). We define the stream function ψ of ϕ by
ψ(x) = Z
R2
(x − y)⊥
2π|x − y|2 · ϕ(y)dy.
It is well-known that ψ ∈ Cb1(R+; C∞) and ∇⊥ψ = ϕ. The stream function ψ given
above is characterized by two facts. One is that ∇⊥ψ = ϕ and another one is that it vanishes at infinity. But in our case, the vanishing at infinity is not important since we will use compactly supported test functions. On the other hand, it is useful to have the stream function small in the neighborhood of the rigid body. We define now a modified stream function, denoted by ψε, which vanishes at the center of the disk D(hε(t), ε):
(14) ψε(t, x) = ψ(t, x) − ψ(t, hε(t)).
Observe that even if ϕ is constant in time, the modified stream function still depends on the time through hε. We collect some properties of the modified stream function in the
following lemma.
Lemma 1. The modified stream function ψε has the following properties:
(i) We have that ψε∈ W1,∞(R+; C∞) and ∇⊥ψε= ϕ.
(ii) For all t, R ≥ 0 and x ∈ R2 we have that kψε(t, ·)kL∞(D(h
ε(t),R))≤ Rkϕ(t, ·)kL∞
(15)
and k∂tψε(t, ·)kL∞(D(h ε(t),R))≤ Rk∂tϕ(t, ·)kL∞+ |h 0 ε(t)|kϕ(t, ·)kL∞ (16)
with the remark that the last relation holds true only almost everywhere in time. Proof. Clearly ∇⊥ψε = ∇⊥ψ = ϕ. Since hε ∈ W1,∞(R+) and ψ ∈ Cb1(R+; C∞) we
immediately see that ψε∈ W1,∞(R+; C∞) which proves (i).
By the mean value theorem
(17) |ψε(t, x)| = |ψ(t, x) − ψ(t, hε(t))| ≤ |x − hε(t)|k∇ψ(t, ·)kL∞ = |x − hε(t)|kϕ(t, ·)kL∞.
Relation (15) follows. To prove (16) we recall that hε is Lipschitz in time so it is almost
everywhere differentiable in time. Let t be a time where hε is differentiable. We write
∂tψε(t, x) = ∂t(ψ(t, x) − ψ(t, hε(t))) = ∂tψ(t, x) − ∂tψ(t, hε(t)) − h0ε(t) · ∇ψ(t, hε(t)) so k∂tψε(t, ·)kL∞(D(h ε(t),R))≤ k∂tψ(t, x) − ∂tψ(t, hε(t))kL∞(D(hε(t),R))+ |h 0 ε(t)|k∇ψ(t, ·)kL∞ ≤ Rk∂t∇ψ(t, ·)kL∞+ |h0ε(t)|kϕ(t, ·)kL∞ = Rk∂tϕ(t, ·)kL∞+ |h0ε(t)|kϕ(t, ·)kL∞.
This completes the proof of the lemma.
We will need to define a cut-off function near the rigid body with L2 norm of the gradient as small as possible. This will be done in the next section. For the moment, let us recall that the function that minimizes the L2 norm of the gradient, that vanishes for |x| = A and is equal to 1 for |x| = B is harmonic. So it is given by the explicit formula
fA,B: R2 → [0, 1], fA,B(x) = 0 if |x| < A ln |x|−ln A ln B−ln A if A < |x| < B 1 if |x| > B.
This special cut-off has the following properties. Lemma 2. We have that fA,B ∈ W1,∞. Moreover,
kfA,B(x) − 1k2L2 = πA2 α2 2 ln2α − 1 2 ln2α − 1 ln α, k∇fA,Bk2L2 = 2π ln α and |x|∇2fA,B 2 L2(A<|x|<B)= 4π ln α where α = BA.
Proof. The Lipschitz character of fA,B is obvious once we remark that fA,B is smooth for
|x| 6= A and |x| 6= B and continuous across |x| = A and |x| = B.
Next, we have that kfA,B(x) − 1k2L2 = Z |x|<A 1 dx + Z A<|x|<B ln |x| − ln B ln B − ln A 2 dx = πA2+ B 2 (ln B − ln A)2 Z A/B<|y|<1 ln2|y| dy = πA2 1 + α 2 ln2α Z 1 1/α ln2r 2r dr = πA2 α 2 2 ln2α − 1 2 ln2α − 1 ln α. From the definition of fA,B, we compute for A < |x| < B
∇fA,B = x |x|2ln α and |∇ 2f A,B| = √ 2 |x|2ln α· So k∇fA,Bk2L2 = 1 ln2α Z A<|x|<B 1 |x|2dx = 2π ln α and |x|∇2fA,B 2 L2(A<|x|<B)= 2 ln2α Z A<|x|<B 1 |x|2dx = 4π ln α.
This completes the proof of the lemma.
3. Cut-off near the rigid body
We begin now the proof of Theorem 4. It suffices to prove the following statement. Proposition 1. For all finite times T > 0 there exists a subsequence vεk which converges
weak∗ in L∞(0, T ; L2) ∩ L2(0, T ; H1) towards a solution v ∈ L∞(0, T ; L2) ∩ L2(0, T ; H1) of the Navier-Stokes equations on [0, T ) × R2 with initial data v0.
Indeed, let us assume that Proposition 1 is proved. We know that the Navier-Stokes equations in dimension two have a unique global solution v in the space L∞(R+; L2) ∩
L2(R+; H1), see for example [12]. The solution v from Proposition 1 is necessarily the
restriction to [0, T ] of this unique global solution. Since we have uniqueness of the limit, we deduce that the whole sequence vε converges weak∗ in L∞(0, T ; L2) ∩ L2(0, T ; H1)
towards v. Since T is arbitrary, Theorem 4 follows.
The rest of this paper is devoted to the proof of Proposition 1. Let T > 0 be fixed. From now on the time t is assumed to belong to the interval [0, T ]. The constant K will denote a constant which depends only on ν and
sup
0<ε≤1
kvεkL∞(0,T ;L2)∩L2(0,T ;H1)
and whose value may change from one line to another. In particular, the constant K does not depend on ε.
By hypothesis we know that
(18) lim
ε→0[0,T ]supε|h 0
We assume that ε ≤ 1/100 and we choose αε such that
(19) 100 ≤ αε≤
1
ε, ε→0limαε= ∞ and ε→0limεαε(1 + |h 0
ε(t)|) = 0
uniformly in t ∈ [0, T ]. The existence of such an αε follows from (18). Indeed, we could
choose for instance
αε= max 100, 1 sup [0,T ] pε + ε|h0 ε(t)| .
We construct in the following lemma a special cut-off function fεnear the disk D(hε(t), ε)
such that fε(x) = 0 for all |x| ≤ ε and fε(x) = 1 for all |x| ≥ εαε.
Lemma 3. There exists a smooth cut-off function fε∈ C∞(R2; [0, 1]) such that
(i) fε vanishes in the neighborhood of the disk D(0, ε) and fε = 1 for |x| ≥ εαε;
(ii) there exists a universal constant C such that kfεkL∞ = 1, k∇fεkL2 ≤ C √ ln αε , |x|∇2fε L2 ≤ C √ ln αε and kfε− 1kL2 ≤ C εαε ln αε · Proof. From Lemma 2 we observe that the function
e fε= fε,εαε = 0 if |x| < ε ln(|x|/ε) ln αε if ε < |x| < εαε 1 if |x| > εαε satisfies k efεkL∞ = 1, k∇ efεkL2 ≤ C √ ln αε , |x|∇2feε L2(ε<|x|<εα ε)≤ C √ ln αε and k efε− 1kL2 ≤ C εαε ln αε
so it has all the required properties except smoothness. More precisely, efε is not smooth
across |x| = ε and |x| = εαε. To obtain a smooth function fε from efε we need to cut-off
in the neighborhood of these two circles.
Let g ∈ C0∞(R2; [0, 1]) be such that g(x) = 0 for |x| < 2 and g(x) = 1 for |x| > 4. We define gε1(x) = g x ε = ( 0, |x| < 2ε 1, |x| > 4ε and gε2(x) = 1 − g 8x εαε = ( 1, |x| < εαε 4 0, |x| > εαε 2 .
With the help of all the auxiliary functions above, we define a new function
Clearly fε satisfies (i) and is smooth across |x| = ε and |x| = εαε, so it remains to prove
(ii). From the definition of fε, we immediately see that kfεkL∞ = 1. To simplify the
write-up, we use the notation Lp(a, b) = Lp(a < |x| < b). Clearly g1ε and gε2 are uniformly bounded in L∞and ∇g1
ε and ∇gε2 are uniformly bounded in L2. Using these observations
we estimate k∇fεkL2 ≤ k∇ g1εfeεkL2(2ε,4ε)+ k∇ efεkL2(4ε,εαε 4 )+ k∇ g 2 ε feε− 1kL2(εαε 4 , εαε 2 ) ≤ C k∇ efεkL2 + k efεkL∞(2ε,4ε)+ k efε− 1kL∞(εαε 4 , εαε 2 ) ≤ √C ln αε + C ln αε ≤ √C ln αε
where we used the bounds
k efεkL∞(2ε,4ε)= ln(|x|/ε) ln αε L∞(2ε,4ε)≤ C ln αε and k efε− 1kL∞(εαε 4 , εαε 2 )= ln(|x|/(εαε)) ln αε L∞(εαε 4 , εαε 2 ) ≤ C ln αε . Similarly, using in addition that |x|∇gi
ε L∞ and |x|∇2gi ε
L2 are bounded
indepen-dently of ε for i = 1, 2, we can estimate |x|∇2f ε L2 ≤ |x|∇2 g1 εfeε L2(2ε,4ε)+ |x|∇2 e fε L2(4ε,εαε 4 ) + |x|∇2 g2ε feε− 1 L2(εαε 4 , εαε 2 ) ≤ C |x|∇2feε L2 + k∇ efεkL2+ k efεkL∞(2ε,4ε)+ k efε− 1kL∞(εαε 4 , εαε 2 ) ≤ √C ln αε + C ln αε ≤ √C ln αε . Finally, kfε− 1kL2 ≤ kgε1feε− 1kL2(2ε,4ε)+ k efε− 1kL2(4ε,εαε 4 )+ kg 2 ε feε− 1kL2(εαε 4 , εαε 2 )+ k1kL2(|x|<2ε) ≤ C k efε− 1kL2 + k1kL2(|x|<4ε) ≤ C εαε ln αε + ε ≤ C εαε ln αε .
This completes the proof of the lemma.
The function fε is a cut-off in the neighborhood of the disk D(0, ε). We define now a
cut-off in the neighborhood of the disk D(hε(t), ε) by setting
ηε(t, x) = fε(x − hε(t)).
Lemma 3 immediately implies that ηε has the following properties:
Lemma 4. We have that
(i) ηε∈ W1,∞(R+; C0,σ∞);
(ii) ηε vanishes in the neighborhood of the disk D(hε(t), ε) and ηε= 1 for |x − hε(t)| ≥
εαε;
(iii) there exists a universal constant C such that kηεkL∞ = 1, k∇ηεkL2 ≤ C √ ln αε , |x − hε(t)|∇2ηε L2 ≤ C √ ln αε (20) and kηε− 1kL2 ≤ C εαε ln αε · (21)
Given a test function ϕ ∈ Cb1(R+; C0,σ∞) we construct a test function ϕε on the set
|x − hε(t)| > ε by setting
(22) ϕε= ∇⊥(ηεψε)
where ψε was defined in Section 2 (see relation (14)). We state some properties of ϕε in
the following lemma:
Lemma 5. The test function ϕε has the following properties:
(i) ϕε∈ W1,∞(R+; C0,σ∞) and is supported in the set |x − hε(t)| > ε;
(ii) ϕε→ ϕ strongly in L∞(0, T ; H1) as ε → 0;
(iii) there exists a universal constant C such that
(23) kϕεkL∞(0,T ;H1) ≤ CkϕkL∞(0,T ;H3).
Proof. Since ηε and ψε are W1,∞ in time and smooth in space, so is ϕε. The compact
support in x of ϕε in the set |x − hε(t)| > ε follows from the compact support of ϕ and
the localization properties of ηε. Obviously ϕε is also divergence free so claim (i) follows.
Recalling that ∇⊥ψε= ϕ we write
ϕε− ϕ = ∇⊥(ηεψε) − ϕ = ∇⊥ηεψε+ ηε∇⊥ψε− ϕ = ∇⊥ηεψε+ (ηε− 1)ϕ.
Using the bound (15) and recalling that ∇ηε is supported in D(hε(t), εαε) we can
estimate kϕε− ϕkL2 ≤ k(ηε− 1)ϕkL2+ k∇ηεψεkL2 ≤ kηε− 1kL2kϕkL∞+ k∇ηεkL2kψεkL∞(D(h ε(t),εαε)) ≤ kηε− 1kL2kϕkL∞+ εαεk∇ηεkL2kϕkL∞ = kϕkL∞(kηε− 1kL2+ εαεk∇ηεkL2).
Taking the supremum on [0, T ] and using (19), (20) and (21) we deduce that
(24) kϕε− ϕkL∞(0,T ;L2)≤ C εαε √ ln αε kϕkL∞([0,T ]×R2)−→ 0.ε→0 Next, k∇(ϕε− ϕ)kL2 = k∇∇⊥ (ηε− 1)ψεkL2 ≤ k∇∇⊥ηεψεkL2 + Ck∇ηεkL2k∇ψεkL∞+ kηε− 1kL2k∇2ψεkL∞.
Recalling that ∇⊥ψε= ϕ and using again Lemma 4 we infer that k∇(ϕε− ϕ)kL2 ≤ C √ ln αε kϕkL∞+ Ck∇ηεkL2kϕkL∞+ kηε− 1kL2k∇ϕkL∞ ≤ √C ln αε kϕkW1,∞.
Combining this bound with (24) implies that
kϕε− ϕkL∞(0,T ;H1)≤
C √
ln αε
kϕkL∞(0,T ;W1,∞)−→ 0.ε→0
In addition, we obtain that there exists a universal constant C > 0 such that
kϕεkL∞(0,T ;H1)≤ CkϕkL∞(0,T ;H1∩W1,∞).
Using the Sobolev embedding H3 ,→ W1,∞ completes the proof of the lemma.
We end this section with an estimate on the H−1 norm of the time-derivative of ϕε.
Lemma 6. Let w be an H1 vector field. There exists a universal constant C > 0 such that for all times t ≥ 0 where hε is differentiable we have that
Z R2 w(x) · ∂tϕε(t, x) − ∂tϕ(t, x)dx ≤ Ck curl wkL2 ε2α2ε ln αε k∂tϕ(t, ·)kL∞ +√εαε ln αε |h0ε(t)|kϕ(t, ·)kL∞.
Proof. Let t be a time where hε is differentiable. We use (22) to write
Z R2 w(x) · ∂tϕε(t, x) − ∂tϕ(t, x)dx = Z R2 w · ∂t∇⊥ (ηε− 1)ψε = − Z R2 curl w ∂t (ηε− 1)ψε = − Z R2 curl w ∂tηεψε− Z R2 curl w (ηε− 1)∂tψε. Clearly ∂tηε= ∂t(fε(x − hε(t))) = −h0ε(t) · ∇fε(x − hε(t))
is supported in the set {|x − hε(t)| ≤ εαε}. We can therefore bound
Z R2 curl w ∂tηεψε ≤ C|h 0 ε(t)| Z |x−hε(t)|≤εαε | curl w||∇fε(x − hε(t))||ψε| ≤ C|h0ε(t)|k curl wkL2k∇fεkL2kψεkL∞(D(h ε(t),εαε)) ≤ C√εαε ln αε |h0ε(t)|k curl wkL2kϕkL∞
where we used (15) and Lemma 3.
Similarly, ηε− 1 is supported in the set {|x − hε(t)| ≤ εαε} so we can use (16) and (21)
to deduce that Z R2 curl w (ηε− 1)∂tψε ≤ k curl wkL2kηε− 1kL2k∂tψεkL∞(D(h ε(t),εαε)) ≤ C εαε ln αε k curl wkL2(εαεk∂tϕkL∞+ |h0 ε(t)|kϕkL∞).
The conclusion follows putting together the above relations.
4. Temporal estimate and strong convergence
The aim of this section is to prove the strong convergence of some sub-sequence of vε.
More precisely, we will prove the following result.
Lemma 7. There exists a sub-sequence vεk of vε which converges strongly in L
2(0, T ; L2 loc).
To prove this lemma we first show some time-derivative estimates and then use the Ascoli theorem.
Let ϕ ∈ C0,σ∞(R2) be a test function which does not depend on the time. Even though ϕ does not depend on t, we can still perform the construction of the cut-off ϕε as in Section
3 (see relation (22)) and all the results of that section remain valid. Observe that even though ϕ does not depend on the time, the modified test function ϕε is time-dependent.
Let us denote by Hσs the space of Hs divergence free vector fields on R2. We endow Hσs with the Hs norm. The dual space of Hσs is Hσ−s. We have that C0,σ∞ is dense in Hσs for all s ∈ R.
Let t ∈ [0, T ] be fixed. We use Lemma 4 and relation (15) to bound Z R2 vε(t, x) · ϕε(t, x) dx = Z R2 vε· ∇⊥(ηεψε) dx = Z R2 vε· (∇⊥ηεψε+ ηεϕ) dx ≤ kvεkL2k∇ηεkL2kψεkL∞(D(h ε(t),εαε))+ kvεkL2kηεkL∞kϕkL2 ≤ C√εαε ln αε kvεkL2kϕkL∞+ kvεkL2kϕkL2 ≤ K1kϕkH2
for some constant K1 independent of ε and t. We used above the Sobolev embedding
H2 ,→ L∞, the boundedness of vε in L∞(0, T ; L2) and relations (19) and (20). We infer
that, for fixed t, the map
C0,σ∞ 3 ϕ 7→ Z
R2
vε(t, x) · ϕε(t, x) dx ∈ R
is linear and continuous for the H2 norm. Since the closure of C∞
0,σ for the H2 norm is Hσ2,
the above map can be uniquely extended to a continuous linear mapping from Hσ2 to R. Therefore it can be identified to an element of the dual of Hσ2 which is Hσ−2. We conclude that there exists some Ξε(t) ∈ Hσ−2 such that
hΞε(t), ϕi =
Z
R2
vε(t, x) · ϕε(t, x) dx ∀ϕ ∈ Hσ2.
Above h·, ·i denotes the duality bracket between Hσ−2 and Hσ2 which is the extension of the usual L2 scalar product. In addition, we have that kΞε(t)kH−2 ≤ K1, so Ξεbelongs to
the space L∞(0, T ; Hσ−2) and is bounded independently of ε in this space.
Because ϕε is compactly supported in {|x − hε(t)| > ε} it can be used as test function
in (13). Multiplying (13) by ϕε and integrating in space and time from s to t yields
We integrate by parts in time the first term above: Z t s Z R2 ∂τvε· ϕε= Z R2 vε(t, x) · ϕε(t, x) dx − Z R2 vε(s, x) · ϕε(s, x) dx − Z t s Z R2 vε· ∂τϕε = hΞε(t) − Ξε(s), ϕi − Z t s Z R2 vε· ∂τϕε. We deduce that (25) hΞε(t) − Ξε(s), ϕi = Z t s Z R2 vε· ∂τϕε− ν Z t s Z R2 ∇vε: ∇ϕε− Z t s Z R2 vε· ∇vε· ϕε. We bound first ν Z t s Z R2 ∇vε: ∇ϕε ≤ ν Z t s k∇vεkL2k∇ϕεkL2 ≤ Cν(t − s)12kϕk H3k∇vεkL2([0,T ]×R2) ≤ K(t − s)12kϕkH3
where we used (23) and the hypothesis that vε is bounded in L2(0, T ; H1).
To estimate the last term in (25) we use the Gagliardo-Nirenberg inequality kf kL4 ≤
Ckf k 1 2 L2k∇f k 1 2
L2, the boundedness of vε in the space displayed in (12) and relation (23):
Z t s Z R2 vε· ∇vε· ϕε = | Z t s Z R2 vε· ∇ϕε· vε ≤ Z t s kvεk2L4k∇ϕεkL2 ≤ Z t s kvεkL2k∇vεkL2kϕεkH1 ≤ C(t − s)12kvεkL∞(0,T ;L2)k∇vεkL2([0,T ]×R2)kϕkH3 ≤ K(t − s)12kϕk H3.
It remains to estimate the first term on the right-hand side of (25). To do that, we use Lemma 6. Recalling that ϕ does not depend on the time, we can write
Z t s Z R2 vε· ∂τϕε ≤ C εαε √ ln αε Z t s k curl vεkL2|h0ε|kϕkL∞ ≤ C√εαε ln αε kϕkH2 Z t s k curl vεkL2|h0ε| ≤ C(t − s)12kϕkH2 εαε √ ln αε sup [0,T ] |h0ε|k curl vεkL2([0,T ]×R2).
Due to the hypothesis imposed on αε, see (19), we know that √εαln αε
ε sup[0,T ]|h
0
ε| goes to
0 as ε → 0. In particular it is bounded uniformly in ε.
Recalling again the boundedness of vεin the space L2(0, T ; H1), we infer from the above
relations that hΞε(t) − Ξε(s), ϕi ≤ K(t − s) 1 2kϕk H3
where the constant K does not depend on ε and ϕ. By density of C0,σ∞ in Hσ3 we infer that kΞε(t) − Ξε(s)kH−3 ≤ K(t − s)
1
2. The functions Ξε(t) are therefore equicontinuous
in time with values in Hσ−3. They are also bounded in Hσ−3 because we already know
that they are bounded in Hσ−2. Since the embedding H−3,→ Hloc−4 is compact, the Ascoli theorem implies that there exists a subsequence Ξεk of Ξε which converges strongly in
C0([0, T ]; Hloc−4).
Recalling the definition of Ξε and using Lemma 4 we can write
|hΞε(t) − vε(t), ϕi| = Z R2 vε· (∇⊥ηεψε+ ηεϕ) dx − Z R2 vε· ϕ = Z R2 vε· (∇⊥ηεψε+ (ηε− 1)ϕ) dx ≤ CkvεkL2k∇ηεkL2kψεkL∞(D(h ε(t),εαε))+ CkvεkL2kηε− 1kL2kϕkL∞ ≤ CkvεkL2kϕkL∞ √εαε ln αε + εαε ln αε ≤ CkvεkL2kϕkH2 εαε √ ln αε . Hence kΞε(t) − vε(t)kH−2 ≤ CkvεkL2 εαε √ ln αε ε→0 −→ 0 uniformly in time. Recalling that Ξεk converges strongly in H
−4
loc uniformly in time we
infer that vεk also converges strongly in L
∞(0, T ; H−4
loc). The interpolation inequality
k · kL2 ≤ k · k 1 5 H−4k · k 4 5
H1 and the boundedness of vε in L2(0, T ; H1) finally imply that vεk
converges strongly in L52(0, T ; L2
loc) ,→ L2(0, T ; L2loc). This completes the proof of Lemma
7.
5. Passing to the limit
In this section we complete the proof of Theorem 4. It is now only a matter of putting together the results proved in the previous sections.
Given the boundedness of vεin L∞(0, T ; L2) ∩ L2(0, T ; H1) and Lemma 7, we know that
there exists some v ∈ L∞(0, T ; L2) ∩ L2(0, T ; H1) and some sub-sequence vεk such that
vεk * v weak∗ in L ∞(0, T ; L2) vεk * v weakly in L 2(0, T ; H1) (26) and vεk → v strongly in L 2(0, T ; L2 loc). (27)
Let ϕ ∈ C0∞([0, T )×R2) be a divergence-free vector field. We construct ϕεk as in Section
3, see relation (22). Since ϕεk is compactly supported in the set {|x − hεk(t)| > εk}, we
can use it as test function in (13) written for εk. We multiply (13) by ϕεk and integrate
by parts in time and space to obtain that (28) − Z T 0 Z R2 vεk· ∂tϕεk+ ν Z T 0 Z R2 ∇vεk : ∇ϕεk + Z T 0 Z R2 vεk · ∇vεk· ϕεk = Z R2 vεk(0) · ϕεk(0).
First, we know by hypothesis that vεk(0) * v0 weakly in L
2. From Lemma 5 we also
have that ϕεk(0) → ϕ(0) strongly in L
2, so (29) Z R2 vεk(0) · ϕεk(0) εk→0 −→ Z R2 v(0) · ϕ(0). Next, we also know from Lemma 5 that ∇ϕεk → ∇ϕ strongly in L
2([0, T ] × R2). Given
that ∇vεk * ∇v weakly in L
2([0, T ] × R2), see relation (26), we infer that
(30) Z T 0 Z R2 ∇vεk : ∇ϕεk ε−→k→0 Z T 0 Z R2 ∇v : ∇ϕ.
The nonlinear term also passes to the limit quite easily. We decompose Z T 0 Z R2 vεk · ∇vεk· ϕεk = Z T 0 Z R2 vεk· ∇vεk· ϕ + Z T 0 Z R2 vεk· ∇vεk· (ϕεk − ϕ).
We know from (27) that vεk → v strongly in L
2(0, T ; L2
loc), from (26) that ∇vεk * ∇v
weakly in L2(0, T ; L2). Recalling that ϕ is compactly supported and since we obviously have that ϕ is uniformly bounded in space and time we can pass to the limit in the first term on the right-hand side:
Z T 0 Z R2 vεk· ∇vεk · ϕ εk→0 −→ Z T 0 Z R2 v · ∇v · ϕ.
To pass to the limit in the second term we make an integration by parts and use the H¨older inequality, the Gagliardo-Nirenberg inequality kf kL4 ≤ Ckf k
1 2 L2k∇f k 1 2 L2 and Lemma 5 Z T 0 Z R2 vεk · ∇vεk· (ϕεk− ϕ) = Z T 0 Z R2 vεk⊗ vεk : ∇(ϕεk − ϕ) ≤ Z T 0 kvεkk 2 L4k∇(ϕεk− ϕ)kL2 ≤ C Z T 0 kvεkkL2k∇vεkkL2k∇(ϕεk − ϕ)kL2 ≤ CT12kvε kkL∞(0,T ;L2)kvεkkL2(0,T ;H1)kϕεk − ϕkL∞(0,T ;H1) ε−→ 0.k→0 We infer that (31) Z T 0 Z R2 vεk· ∇vεk· ϕεk εk→0 −→ Z T 0 Z R2 v · ∇v · ϕ.
The last term we need to pass to the limit is the term with the time-derivative. Thanks to Lemma 6 we can bound
where we used (19). But we also have that Z T 0 Z R2 vεk· ∂tϕ εk→0 −→ Z T 0 Z R2 v · ∂tϕ
so we can conclude that (32) Z T 0 Z R2 vεk∂tϕεk εk→0 −→ Z T 0 Z R2 v · ∂tϕ.
Gathering (28), (29), (30), (31) and (32), we conclude that − Z T 0 Z R2 v · ∂tϕ + ν Z T 0 Z R2 ∇v : ∇ϕ + Z T 0 Z R2 v · ∇v · ϕ = Z R2 v(0) · ϕ(0)
which is the weak formulation of Navier-Stokes equations in R2. This completes the proof of Proposition 1.
Acknowledgments
J.H. and D.I. have been partially funded by the ANR project Dyficolti ANR-13-BS01-0003-01. D.I. has been partially funded by the LABEX MILYON (ANR-10-LABX-0070) of Universit´e de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).
References
[1] M. Chipot, G. Planas, J. C. Robinson, and W. Xue. Limits of the Stokes and Navier-Stokes equations in a punctured periodic domain. arXiv:1407.6942 [math], 2014.
[2] B. Desjardins and M. J. Esteban. Existence of weak solutions for the motion of rigid bodies in a viscous fluid. Archive for Rational Mechanics and Analysis, 146(1):59–71, 1999.
[3] S. Ervedoza, M. Hillairet, and C. Lacave. Long-Time Behavior for the Two-Dimensional Motion of a Disk in a Viscous Fluid. Communications in Mathematical Physics, 329(1):325–382, 2014.
[4] O. Glass, C. Lacave, and F. Sueur. On the motion of a small body immersed in a two dimensional incompressible perfect fluid. Bulletin de la Soci´et´e Math´ematique de France, 142(3):489–536, 2014. [5] O. Glass, C. Lacave, and F. Sueur. On the Motion of a Small Light Body Immersed in a Two
Dimensional Incompressible Perfect Fluid with Vorticity. Communications in Mathematical Physics, 341(3):1015–1065, 2016.
[6] O. Glass, A. Munnier, and F. Sueur. Point vortex dynamics as zero-radius limit of the motion of a rigid body in an irrotational fluid. Inventiones mathematicae, 214(1):171–287, 2018.
[7] K.-H. Hoffmann and V. N. Starovoitov. On a motion of a solid body in a viscous fluid. Two-dimensional case. Advances in Mathematical Sciences and Applications, 9(2):633–648, 1999.
[8] D. Iftimie, M. C. Lopes Filho, and H. J. Nussenzveig Lopes. Two Dimensional Incompressible Ideal Flow Around a Small Obstacle. Communications in Partial Differential Equations, 28(1-2):349–379, 2003.
[9] D. Iftimie, M. C. Lopes Filho, and H. J. Nussenzveig Lopes. Two-dimensional incompressible viscous flow around a small obstacle. Mathematische Annalen, 336(2):449–489, 2006.
[10] C. Lacave. Two-dimensional incompressible viscous flow around a thin obstacle tending to a curve. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 139(6):1237–1254, 2009. [11] C. Lacave and T. Takahashi. Small Moving Rigid Body into a Viscous Incompressible Fluid. Archive
for Rational Mechanics and Analysis, 223(3):1307–1335, 2017.
[12] J.-L. Lions. Quelques m´ethodes de r´esolution des probl`emes aux limites non lin´eaires. Dunod, 1969. [13] J. San Mart´ın, V. Starovoitov, and M. Tucsnak. Global Weak Solutions for the Two-Dimensional
Motion of Several Rigid Bodies in an Incompressible Viscous Fluid. Archive for Rational Mechanics and Analysis, 161(2):113–147, 2002.
[14] D. Serre. Chute Libre d’un Solide dans un Fluide Visqueux lncompressible. Existence. Japan Journal of Industrial and Applied Mathematics, 4(1):99–110, 1987.
[15] T. Takahashi and M. Tucsnak. Global strong solutions for the two-dimensional motion of an infinite cylinder in a viscous fluid. Journal of Mathematical Fluid Mechanics, 6(1):53–77, 2004.
[16] N. V. Yudakov. The solvability of the problem of the motion of a rigid body in a viscous incompressible fluid. Dinamika Sploshnoi Sredy, 18:249–253, 1974.
J. He: Universit´e de Lyon, Universit´e Lyon 1 – CNRS UMR 5208 Institut Camille Jordan – 43 bd. du 11 Novembre 1918 – Villeurbanne Cedex F-69622, France. Email: jiao.he@math.univ-lyon1.fr
D. Iftimie: Universit´e de Lyon, Universit´e Lyon 1 – CNRS UMR 5208 Institut Camille Jordan – 43 bd. du 11 Novembre 1918 – Villeurbanne Cedex F-69622, France.
Email: iftimie@math.univ-lyon1.fr
